Results 1 
2 of
2
The bead model & limit behaviors of dimer models
"... In this paper, we study the bead model: beads are threaded on a set of wires on the plane represented by parallel straight lines. We add the constraint that between two consecutive beads on a wire, there must be exactly one bead on each neighboring wire. We construct a oneparameter family of Gibbs ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
(Show Context)
In this paper, we study the bead model: beads are threaded on a set of wires on the plane represented by parallel straight lines. We add the constraint that between two consecutive beads on a wire, there must be exactly one bead on each neighboring wire. We construct a oneparameter family of Gibbs measures on the bead configurations that are uniform in a certain sense. When endowed with one of these measures, this model is shown to be a determinantal point process, whose marginal on each wire is the sine process (given by eigenvalues of large hermitian random matrices). We prove then that this process appears as a limit of any dimer model on a planar bipartite graph when some weights degenerate. 1 Introduction and presentation of the bead model We consider the collection of configurations of beads strung on an infinite collection of parallel threads lying on the plane. A bead configuration on these threads gives a configuration of points on Z × R. We impose the following constraints on the configurations: Figure 1: A bead configuration.
The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions
 Mem. Amer. Math. Soc
"... Abstract. We define the correlation of holes on the triangular lattice under periodic boundary conditions and study its asymptotics as the distances between the holes grow to infinity. We prove that the joint correlation of an arbitrary collection of latticetriangular holes of even sides satisfies, ..."
Abstract

Cited by 12 (10 self)
 Add to MetaCart
(Show Context)
Abstract. We define the correlation of holes on the triangular lattice under periodic boundary conditions and study its asymptotics as the distances between the holes grow to infinity. We prove that the joint correlation of an arbitrary collection of latticetriangular holes of even sides satisfies, for large separations between the holes, a Coulomb law and a superposition principle that perfectly parallel the laws of two dimensional electrostatics, with physical charges corresponding to holes, and their magnitude to the difference between the number of rightpointing and leftpointing unit triangles in each hole. We detail this parallel by indicating that, as a consequence of our result, the relative probabilities of finding a fixed collection of holes at given mutual distances (when sampling uniformly at random over all unit rhombus tilings of the complement of the holes) approaches, for large separations between the holes, the relative probabilities of finding the corresponding two dimensional physical system of charges at given mutual distances. Physical temperature corresponds to a parameter refining the background triangular lattice. We give an equivalent phrasing of our result in terms of covering surfaces of given holonomy. From this perspective, two dimensional electrostatics arises by averaging over all possible discrete geometries of the covering surfaces.